My apologies if this has been asked before. I wasn't able to find an explanation that made sense to me on Google.

Suppose we have the equation below, which is written in Einstein notation:

$$y_{j} = x_iz_ix_j$$

and say that both $j$ and $i$ take on values of $1$ and $2$. Is this notation to be interpreted as

$$y_1 = (x_1z_1+x_2z_2)x_1$$ $$y_2 = (x_1z_1+x_2z_2)x_2$$

or as $$y_1+y_2 = (x_1z_1+x_2z_2)(x_1+x_2)?$$

In other words, does the index across the equals sign imply multiple equations, or adding up terms on each side of a single equation?

Context: I am an undergrad math major studying fluid flow. The conservation of momentum equation was written in Einstein Notation and I am having trouble understanding the meaning.

  • $\begingroup$ It's the first. Think about the simpler case $x_i = y_i$. This should mean that the vectors $x$ and $y$ are equal, not that their components sum to the same number. $\endgroup$
    – knzhou
    Feb 21, 2017 at 0:28

1 Answer 1


The correct interpretation is the first one: $$ y_j=x_iz_ix_j$$ Means that to obtain the jth component of $y$ you have to sum over the repeated indeces, i.e.

$$ y_j=\sum_{i=1}^2 x_iz_ix_j$$

  • $\begingroup$ Is it a coincidence that the two interpretations yield the same result in this case, or does it happen that way every time? I guess all we know is true is that the first implies the second. $\endgroup$ Feb 21, 2017 at 0:28
  • $\begingroup$ It's coincidence. You can think at the definition of cross product using the Einstein notation as an example. $\endgroup$ Feb 21, 2017 at 0:34

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